Transcript Document
IT Systems
Number Operations
EN230-1
Justin Champion
C208 – 3273
www.staffs.ac.uk/personel/engineering_and_technology/jjc1
IT Systems
Contents
Maths
•
•
•
•
Addition
Subtraction
Multiply
Division
Negative numbers
How it actually works
• Half adder
• Full Adder
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Different numbers systems
Last week we looked at different common number systems
• Hexadecimal
• Binary
• Decimal
These number systems unless we could do something with
them
Computers are good calculations we will look methods of
doing the maths
First job is to convert the numbers into a common format
• Methods of this were discussed last week in lecture 4
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Binary Addition
This works in the same way as the decimal
system
• Add
1
+0
1
• The issue becomes when we have
1
+1
??
• This sum is done in the same way decimal, the
value is written in the column and a carry bit is
then added to the next column
1
+1
10 = Decimal 2
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Binary Addition (continued)
More examples
• As discussed last week ensure that both lines
have the same number of values so
• Becomes
010
+0111011
0000010
+0111011
• Workings
Carries
1
0000010
+0111011
0111101
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Computer Architecture
6811
• Has a special register which stores the carry bit if
the number is bigger than the register you are
working upon
• Register A, B are both 8 Bits
All processors have this register
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Binary Subtraction
The same technique is used as decimal
Try
1
-0
1
111
- 100
011
11000
- 10111
00001
If you are having problems with this concept there is
nothing wrong with converting to decimal carrying
out the maths then converting back to binary
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Binary Multiplication & Division
Special case where multiplication or division are by
a divisor of 2
111
* 10
1110
Dec 7
Dec 2
This could be done by the same method as used for
decimal
• Faster method is to use Binary Bit Shifting
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Binary Bit Shifting
This is moving all of the bits
• Move them towards the big endian for multiplication
• Move them towards the little endian for division
One movement would equal a change by 2
• If you were multiplying by 8
• You would move towards the big endian 4 times
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Multiplication
Try this
If the numbers are NOT multiples of 2
101
* 100
or
1000
/ 100
• You have to do the work the same as the decimal
method
For the remainders we will ignore as this will
end up complicated
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Negative numbers
Obviously it is possible to end up with negative
numbers
These number are represented by different
methods
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Signed Magnitude
One’s Complement
Two’s Complement
Excess
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Signed Magnitude
The big endian is used to indicate if the value is positive or
negative
• 00001100 == +12
• 10001100==-12
One’s Complement
Positive numbers do not change
Negative numbers flip the bits
The big endian as with signed magnitude indicates a positive or
negative value
Flipping the bits
All 1’s become 0 and all 0’s become 1’s
So 0001001 becomes 1110110
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Two’s Complement
First of all this is the same as 1’s complement
The value of 1 is added to the result
Excess 2(m-1) = 8 bits = 27
The maximum value is added to the binary
number
+7 = 10000111
-7 = 01111001
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Whichever system is used it must be check on
the architecture.
If your maths are based upon 2’s complement
and you use 1’s the answers will be very
different
2’s complement is the common used method
• The ease of maths is the reason for this
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Maths with the 2’s complement
12 + -7
12 = 1100
-7 = 1001
1100 Dec 12
+1001 Dec -7
0101 Dec 5
The fifth bit is thrown away and will be stored by the
overflow bit
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Maths on Hexadecimal numbers
As with Binary the same method as decimal is used
7
Dec 7
9
Dec 9
+2
Dec 2
+2
Dec 2
9
Dec 9
B
Dec 11
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How it works ?
So after all of this how does the computer actually add
the numbers
A series of logic gates
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Sum = A . B
Carry = A + B exclusive OR
This will add 2 binary numbers together
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And is commonly referred to as a half adder
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How it works ?
Half adder adds 2 binary values together and stores one
bit in the carry bit
A full adder accepts the carry bit as a input and also has
a carry bit as a output.
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These full adders are then put into sequence connecting to each
other
So for a 8 bit adder we would have 8 of them in a row
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Summary of what we have discussed
Maths
•
•
•
•
Addition
Subtraction
Multiply
Division
Negative numbers
How it actually works
•
•
Half adder
Full Adder